A&A 607, A79 (2017)DOI: 10.1051/0004-6361/201731335c© ESO 2017

Astronomy&Astrophysics

The abundance of ultra-diffuse galaxies from groups to clusters

UDGs are relatively more common in more massive haloes

Remco F. J. van der Burg1, 2, Henk Hoekstra3, Adam Muzzin4, Cristóbal Sifón5, Massimo Viola3,Malcolm N. Bremer6, Sarah Brough7, Simon P. Driver8, 9, Thomas Erben10, Catherine Heymans11,Hendrik Hildebrandt10, Benne W. Holwerda12, Dominik Klaes10, Konrad Kuijken3, Sean McGee13,

Reiko Nakajima10, Nicola Napolitano14, Peder Norberg15, Edward N. Taylor16, and Edwin Valentijn17

1 IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, Francee-mail: remco.van-der-burg@cea.fr

2 Université Paris Diderot, AIM, Sorbonne Paris Cité, CEA, CNRS, 91191 Gif-sur-Yvette, France3 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands4 Department of Physics and Astronomy, York University, 4700 Keele St., Toronto, Ontario MJ3 1P3, Canada5 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA6 Astrophysics Group, School of Physics, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK7 School of Physics, University of New South Wales, NSW 2052, Australia8 International Centre for Radio Astronomy Research (ICRAR), The University of Western Australia, 35 Stirling Highway, Crawley,

WA 6009, Australia9 School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

10 Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany11 Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill,

Edinburgh EH9 3HJ, UK12 Department of Physics and Astronomy, University of Louisville, Louisville, KY 40292, USA13 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK14 INAF–Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy15 ICC & CEA, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK16 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn 3122, Australia17 Kapteyn Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands

Received 8 June 2017 / Accepted 7 August 2017

ABSTRACT

In recent years, many studies have reported substantial populations of large galaxies with low surface brightness in local galaxyclusters. Various theories that aim to explain the presence of such ultra-diffuse galaxies (UDGs) have since been proposed. A keyquestion that will help to distinguish between models is whether UDGs have counterparts in host haloes with lower masses, andif so, what their abundance as a function of halo mass is. We here extend our previous study of UDGs in galaxy clusters to galaxygroups. We measure the abundance of UDGs in 325 spectroscopically selected groups from the Galaxy And Mass Assembly (GAMA)survey. We make use of the overlapping imaging from the ESO Kilo-Degree Survey (KiDS), from which we can identify galaxies withmean surface brightnesses within their effective radii down to ∼25.5 mag arcsec−2 in the r band. We are able to measure a significantoverdensity of UDGs (with sizes reff ≥ 1.5 kpc) in galaxy groups down to M200 = 1012 M, a regime where approximately only onein ten groups contains a UDG that we can detect. We combine measurements of the abundance of UDGs in haloes that cover threeorders of magnitude in halo mass, finding that their numbers scale quite steeply with halo mass: NUDG(R < R200) ∝ M1.11±0.07

200 . Tobetter interpret this, we also measure the mass-richness relation for brighter galaxies down to M∗

r + 2.5 in the same GAMA groups,and find a much shallower relation of NBright(R < R200) ∝ M0.78±0.05

200 . This shows that compared to bright galaxies, UDGs are relativelymore abundant in massive clusters than in groups. We discuss the implications, but it is still unclear whether this difference is relatedto a higher destruction rate of UDGs in groups or if massive haloes have a positive effect on UDG formation.

Key words. galaxies: dwarf – galaxies: formation – galaxies: evolution – galaxies: structure – galaxies: groups: general –galaxies: clusters: general

1. Introduction

While early studies of low surface brightness galaxies dateback several decades (e.g. Impey et al. 1988; Turner et al. 1993;Dalcanton et al. 1997), the discovery of a substantial number ofvery large diffuse galaxies in local clusters has drawn significantattention in recent years. van Dokkum et al. (2015) identified asignificant population of large (reff ≥ 1.5 kpc) galaxies with low

central surface brightness (µ(g, 0) = 24−26 mag arcsec−2) in theComa cluster, and dubbed them ultra-diffuse galaxies (UDGs).

In the two years since, several studies reported popula-tions of UDGs in different galaxy cluster samples. Some ofthe studies focused on the nearby and well-studied Fornax(Muñoz et al. 2015), Coma (Koda et al. 2015; Kadowaki et al.2017) and Virgo (Mihos et al. 2015; Davies et al. 2016) clusters.

Article published by EDP Sciences A79, page 1 of 13

A&A 607, A79 (2017)

Other works studied more distant clusters (van der Burg et al.2016; Román & Trujillo 2017a), including a Frontier-fields clus-ter at z = 0.3 (Janssens et al. 2017). It is clear that UDGs area common phenomenon in galaxy clusters, and their ubiquityhas to be understood in the context of existing galaxy formationmodels.

Several explanations have been proposed in the literatureto explain this class of galaxies. van Dokkum et al. (2015) sug-gested that UDGs are some sort of “failed” galaxy, with a rela-tively massive dark matter halo like the one expected for an L∗galaxy. Such a high mass would make them capable of survivingvery close (down to only ∼300 kpc, see van der Burg et al. 2016)to the cluster centres. They may have stopped forming stars dueto feedback processes, which is again possibly related to a mas-sively overdense environment. Yozin & Bekki (2015) were in-deed able to reproduce several observed properties of UDGs witha simple tidal disruption model that linked their presence exclu-sively to dense environments.

There are also less exotic explanations. Amorisco & Loeb(2016) claimed that UDGs can be readily explained by a standardmodel of disk formation, and that UDGs are simply the extremesof a continuous distribution in size and luminosity. This wouldmean that they also exist in lower-density environments, albeitwith possibly more disk-like morphologies. Rong et al. (2017)reached a similar conclusion based on semi-analytic galaxy for-mation models. Di Cintio et al. (2017) used the NIHAO simula-tion to study the formation of UDG-type galaxies, and they alsosuggested that internal processes, particularly outflow-drivenfeedback, may be responsible for their formation. Feedback fromactive galactic nuclei in dwarf galaxies may also contribute totheir diversity and range in morphologies (Silk 2017).

Such opposing formation scenarios make different, obser-vationally testable, predictions. One way forward is to investi-gate the failed-galaxy scenario by measuring the halo massesof UDGs. Beasley et al. (2016) estimated the mass of a UDGin the Virgo cluster based on the velocity dispersion of glob-ular clusters that are associated with this galaxy, finding thatthey are overly massive for their stellar mass. Several otherstudies also based their mass estimates on the specific fre-quency, or number, of globular clusters in UDGs, reaching con-tradictory conclusions (Amorisco et al. 2016; Peng & Lim 2016;Beasley & Trujillo 2016). van Dokkum et al. (2016) measured avelocity dispersion of UDG DF44, which, upon extrapolation toM200, is consistent with a halo mass of M200 ∼ 1012 M. In con-trast, Sifón et al. (2017) measured the average mass associatedwith UDGs directly using weak gravitational lensing, findingthat this is too low for them all to be “failed” massive galaxies.It is likely that the range of mass measurements of individualUDGs is real, and not (only) due to different assumptions in theanalyses of different studies. This may hint at a range of differentformation scenarios for UDGs (e.g. Zaritsky 2017).

Another critical observational study to constrain the forma-tion scenario of UDGs is to measure their abundance in dif-ferent environments. van der Burg et al. (2016, hereafter vdB16)studied eight clusters that span an order of magnitude in halomass, and found that the number of UDGs per cluster scales asNUDG ∝ M0.93±0.16

200 , where M200 is the virial mass of the host halo.At even lower halo masses, Román & Trujillo (2017b) were ableto identify UDGs in three nearby groups by making use of three-band photometric data from the SDSS deep Stripe 82. Theirwork shows that UDGs are also present in galaxy groups, with anabundance that is close to what is expected from extending thevdB16 relation down to lower masses. We note, however, thatthe sample they studied are the three Hickson Compact Groups

(Hickson 1982) that overlap with the SDSS Stripe 82, and thismay not be fully representative of the underlying population ofgroups.

In this work we perform a systematic search for UDGs ingalaxy groups, considering the 197 deg2 overlap region betweenthe optical-imaging Kilo-Degree Survey (KiDS) with the groupsidentified with the Galaxy And Mass Assembly (GAMA) spec-troscopic survey. The groups are selected completely indepen-dently of their possible UDG content. By performing an analy-sis that is close to the one presented in vdB16, this allows us tostudy the abundance of UDGs in a consistent way over a widehalo mass range. A previous search for galaxies with low sur-face brightness was performed using SDSS imaging of the sameGAMA region (Williams et al. 2016). Based on the locations ofthese galaxies, it was shown that they are plausibly associatedwith the z < 0.1 large-scale structure (Fig. 6 in Williams et al.2016). In this paper we quantify this statement, focusing on theGAMA groups, based on the standardised galaxy selection usingthe UDG criteria on the KiDS imaging.

The paper is organised as follows. Section 2 provides anoverview of the data products from the KiDS and GAMA sur-veys we used. Section 3 describes the selection of our sampleof UDGs, and how we associated them with the galaxy groupsin a statistical sense. Section 4 presents our main results: theabundance of UDGs as a function of halo mass, and the size dis-tribution of the UDG sample. We discuss our findings in Sect. 5,compare them to previous work and to a measurement of themass-richness relation for the same GAMA groups to help to in-terpret our results. We summarise in Sect. 6. Appendix A listsseveral robustness tests that we performed to increase our confi-dence in the presented results.

All magnitudes we quote are in the AB magnitudes system,and we adopt angular-diameter and luminosity distances corre-sponding to ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7 andH0 = 70 km s−1 Mpc−1. Unless stated otherwise, error bars de-note 1σ uncertainties, or 68% confidence intervals.

2. Data

Our UDG candidates are identified using photometric data fromthe KiDS (de Jong et al. 2015), and the environments (in par-ticular group properties) are characterised using the GAMA(Driver et al. 2011) group catalogue. In this study, we focus onthe overlap region between the KiDS and GAMA surveys, to-talling an effective (unmasked) area of 170 square degrees, andthe surveys are summarised in turn below.

2.1. Kilo-Degree Survey

KiDS is an on-going ESO wide-area imaging survey in theugri bands that will cover 1500 deg2, and which is designed tomap the matter distribution in the Universe using weak grav-itational lensing. Here we used the “KiDS-450” data release,which covers approximately 450 deg2 (Hildebrandt et al. 2017;de Jong et al. 2017).

Our starting point is the KiDS r-band imaging, reduced andstacked with the THELI (Erben et al. 2013) pipeline. The dataare taken under dark conditions with minimum atmospheric ex-tinction. To facilitate weak-lensing measurements in the r band,they are taken in conditions with excellent seeing (PSF FWHM <0.8′′). Stacks are composed of five dithers with a combined inte-gration time of 1800 s.

The stacks are designed for optimal weak-lensing shapemeasurements, and there are still residual background patterns

A79, page 2 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Fig. 1. Three KiDS+GAMA fields. The combined KiDS area is shown in white, with regions masking bright stars, reflective haloes, and imageartefacts shown in grey. Circles indicate the actual angular sizes of R200 disks of GAMA groups up to z ≤ 0.10, with colours referring to theirredshifts (cf. colour scale on the right). Small red dots show the positions of all UDG candidates that satisfy our final selection criteria.

that we have to account for in this study. We subtracted theseresidual background effects using a mesh size of 150 pixels(∼32′′), smoothed with a median filter of size 5×5 meshes. Thiscorrects background structures on a scale that is much larger thanthe sizes of the galaxies we study.

We used the conservative masks designed by the KiDS team,which mask ∼15% of the total survey area. This includes brightstars, their reflective haloes, and image artefacts (for more details

see Kuijken et al. 2015). Figure 1 shows the unmasked area ofthe three KiDS fields we consider in this study in white.

2.2. Galaxy And Mass Assembly

GAMA (Driver et al. 2011; Liske et al. 2015) is a spectro-scopic survey that is complete down to an r-band magnitudelimit of 19.8. Each GAMA field is visited several times toovercome sampling problems that are due to close galaxy pairs,

A79, page 3 of 13

A&A 607, A79 (2017)

Table 1. Rough overview of the three KiDS+GAMA fields studied here.

Field KiDS tiles Eff area [deg2] UDG candidates GroupsG09 64 50.7 4635 83G12 64 56.2 5815 140G15 69 62.5 6145 102Total 197 169.5 16595 325

Notes. Each KiDS tile covers roughly 1 deg2. The effective areasare reduced by ∼15% due to the masking, see Fig. 1 for a graphicalrepresentation.

which are common in groups and clusters of galaxies. Usinga friends-of-friends (FoF) algorithm on the spectroscopic cat-alogue, Robotham et al. (2011) have constructed a group cat-alogue for the GAMA survey regions that links ∼14% of theGAMA galaxies to a group with at least five spectroscopic mem-bers. Since this first version of the catalogue (G3Cv1), the link-ing scheme has been updated several times. Including the currentversion of the spectroscopic catalogue, G3Cv9 is now the mostrecent version of the group catalogue. While it covers a total offive survey fields, we focus on the three Equatorial GAMA fieldsG09, G12, and G15, which overlap with KiDS (cf. Table 1). Weconsidered all groups from G3Cv9 in these survey fields thathave at least five FoF members and are in the redshift range0.01 ≤ z ≤ 0.10. This selection resulted in a total of 325 groups,see Table 1.

Several group mass definitions are in use within the GAMAcollaboration. Originally, group masses were measured dynami-cally, using the spectroscopic group members as tracers of thepotential. While this dynamical method is expected to workreasonably well for groups with a large number of members,these masses are unreliable in the low-mass regime, however,where there are few members (Robotham et al. 2011). Instead,the masses used in this work are based on the total group lumi-nosity in the r band. Viola et al. (2015) have performed a weak-lensing study of the overlap region between KiDS and GAMAto measure the total mass associated with groups, in bins of theirtotal r-band luminosity. We used the almost-linear relation theyfound between total luminosity and weak-lensing mass to esti-mate M200 for each group. The R200 is defined, as usual, as theradius corresponding to a sphere that contains this mass. Thegroup centres are defined as their r-band luminosity-weightedcentre of mass.

Figure 1 gives an overview of the three GAMA regions stud-ied here. The white area is the (unmasked) KiDS area consid-ered here. The circles mark the locations of groups, with coloursindicating their redshifts, and their sizes are the projected angu-lar size of their R200 disks. An additional criterion to define ourgroup sample is that less than 50% of their R200 disk is maskedin the KiDS images (which removes 17 groups from the groupsample, bringing the total to 325).

3. Analysis

UDGs are defined in physical units (reff ≥ 1.5 kpc,van Dokkum et al. 2015) and have a maximum size of aboutreff ∼ 7.0 kpc (vdB16). Since we here consider GAMA groupsover a wide redshift range, these physical sizes correspond to arange of different angular sizes. For example, at z = 0.01, wewould have to consider galaxies with angular sizes between 7′′.3

and 34′′, while at z = 0.10, this would correspond to sizes be-tween 0′′.8 and 3′′.8. From an observational point of view, wehave to work in angular units, which initially leads to a redshift-dependent selection. However, when we exploit the wide rangeof redshifts of the GAMA groups, we can make definitive state-ments about their UDG content, as outlined below.

3.1. Large galaxies with low surface brightness in KiDS

Although we work in angular units, we performed an anal-ysis that is as close as possible to the analysis presented invdB16, where data from the Canada-France-Hawaii Telescope(CFHT) were analysed. As a first step, we used SExtractor(Bertin & Arnouts 1996) on the r-band stack to detect sources,using a Gaussian filter with a FWHM of 5 pixels (∼1′′ withour detector) to improve our sensitivity towards faint extendedgalaxies. The KiDS pixel scale is slightly coarser than CFHTMegaCam (0.214′′/pix versus to 0.185′′/pix), so we requiresources to have at least 15 adjacent pixels that are at least0.90 sigma above the background. These choices keep the pu-rity of detecting real objects (and not noise fluctuations) high(cf. vdB16). We detected a total of 12 243 224 sources in theKiDS fields.

We applied liberal selection criteria to these SExtractoroutput catalogues to perform a first selection of large extendedgalaxies (and thus potential UDGs at the redshifts of the GAMAgroups). We selected against stars and other compact objects,requiring that r2′′ > 0.9 + r7′′ , where rx′′ is the r-band magni-tude within a circular diameter of x arcsec. This pre-selectionwas merely intended to speed up the overall analysis. We per-formed several tests to ensure that this cut did not affect the finalselection of galaxies that we aimed to study, even if a fractionof UDGs is nucleated (cf. Yagi et al. 2016). After applying theKiDS survey mask, we were left with 873 702 sources in theconsidered KiDS area.

These sources were our input to GALFIT (Peng et al. 2002),which we used to estimate morphological parameters whilemasking any pixels belonging to neighbouring sources. We thenselected galaxies with best-fit Sérsic parameters in the range24.0 ≤ 〈µ(r, reff)〉 ≤ 26.0 mag arcsec−2, circularised effectiveradii 1′′.5 ≤ reff ≤ 8′′.0 that are fit within 7 pixels from theSExtractor position, and have a Sérsic index n ≤ 4. Wenote that the surface brightness criterion is somewhat shallower(26.0 versus 26.5) than the criterion applied in vdB16 becausewe work with shallower data here. Some examples of selectedgalaxies are shown in Appendix B.

We find that GALFIT provides reasonable fits to the data inmost cases; we therefore trust the GALFIT parameters, and theselection can be considered as being quite pure. We did notmake any additional subjective or by-eye checks for individualsources, so that this study remains reproducible. We assume thatany residual impurity does not correlate with the locations ofgroups and is eventually removed (in a statistical sense) by ourbackground subtraction. Given that KiDS is untargeted, this is areasonable assumption. To test the performance of our procedureand to gauge the completeness of our final UDG candidate selec-tion, we also performed all processing steps on a set of tailoredimage simulations, as described next.

Following vdB16, we injected objects with Sérsic profileswith n = 1 (typical values for UDGs, cf. Koda et al. 2015;van der Burg et al. 2016) at random positions in the KiDS r-bandimage stacks. Effective radii (reff) were drawn uniformly be-tween 1′′ and 10′′ (we worked in angular units here). Centralsurface brightnesses were drawn uniformly from the range

A79, page 4 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Fig. 2. Recovered fraction of sources that were injected in the KiDSr-band stacks as a function of circularised effective radius, and the meansurface brightness within the effective radius. The dotted region definesthe selection criteria used in this study. Arrows show the negligible biasin the morphological parameters recovered with GALFIT in six cornersof parameter space.

22.7 < µ(r, 0)/[mag arcsec−2] < 26.0, and the ellipticity param-eter ε = (1− q)/(1 + q), where q is the ratio of minor/major axis,was drawn uniformly between 0 and 0.2. Injecting 1500 sourcesper realisation, with 10 realisations per KiDS tile, we simulatedseveral million sources.

In the following we work in units of the mean surface bright-ness within the effective radius (following vdB16). This param-eter is exactly 1.12 mag arcsec−2 larger than the central surfacebrightness for Sérsic-profiles with n = 1, and is more closelyrelated to the detectability of a source. In particular, as notedin vdB16, the recovery rate of simulated sources with slightlydifferent Sérsic-indices (n = 0.5 or n = 1.5) is similar whenexpressed in these units. It is also noted there that the detectabil-ity of a source of a given effective (circularized) radius does notsensitively depend on its ellipticity.

Figure 2 shows the completeness of the simulated sources.The completeness is defined as the fraction of injected sources,in bins of size and surface brightness, that are 1) detectedwith SExtractor, 2) pass the pre-selection based on theSExtractor output, and 3) have best-fit morphological param-eters corresponding to the selection criteria we outlined above.These simulated sources were thus processed through exactlythe same pipeline as the “real” sources. While only 2% of “real”sources that were fed to GALFIT satisfy these criteria, the re-covery rate of simulated sources is ∼66%. Most “real” sourcesthat did not match the final selection criteria are either too small(the pre-selection criteria on the SExtractor output catalogueswere very inclusive, as outlined above) or too bright (there wasno pre-selection on brightness).

With our final selection criteria, we are left with 16 595 largeand faint galaxies in an unmasked KiDS area of 169.5 deg2. Weensured that we did not count the same galaxies several timesbecause of the slightly overlapping KiDS tiles. As a sanity check,

we applied exactly the same selection criteria to the sources de-tected in the CFHTLS Deep fields (with noise added to resem-ble the cluster imaging used in vdB16). We found 343 sourcesthat satisfy these criteria (in angular units) in a combined effec-tive area of 3.52 deg2. This means that the source densities areremarkably similar between the two surveys, providing furtherconfidence that we can compare results from these different in-strumental setups and depths.

3.2. Associating UDGs with GAMA groups

We now link the selected sample of galaxies to UDGs at theredshifts of the GAMA groups. As in vdB16, we consideredUDGs with physical sizes between 1.5 and 7.0 kpc. We note thatfor galaxy groups with redshift z > 0.051, we did not probethe smallest UDGs (because they would have an angular sizesmaller than 1′′.5). Correspondingly, for galaxy groups with red-shift z < 0.044, we did not probe the largest UDGs (becausethey have an angular size larger than 8′′.0). As described below,we combined information from all GAMA groups, exploitingthe entire redshift range of the GAMA groups (0.01 ≤ z ≤ 0.10),to turn our samples of galaxies with sizes between 1′′.5 and 8′′.0into physical sizes (and thus UDGs, after accounting for fore-and background interlopers). Given that only 10% of the currentage of the Universe has passed since z = 0.10, we assume thatthe UDG content of the galaxy groups does not evolve over thisredshift range. We proceeded as follows:

– For each GAMA group, we counted the number of largegalaxies with low surface brightness that satisfy our angu-lar selection criteria (cf. Fig. 2), the physical size criteriacorresponding to the group redshift, and that are projectedwithin R200 from the group centre. We estimated the back-ground value on this number, namely the counts expectedfrom the average number density of the particular KiDS fieldstudied here (i.e. G09, G12, or G15)1. The raw counts mi-nus the background expectation and associated Poisson er-rors are our main measurements. The total overdensity ofmeasured UDGs in all GAMA groups is 29% compared tothe background.

– We corrected this number for the masked fraction of the R200disk by assuming that the UDG density is homogeneous. Theaverage correction factor for all groups is 17% to the numbercounts, which is close to the masked fraction of the entireKiDS survey area. We recall that we did not consider groupsin which more than half the R200 disk is masked.

– We corrected this number for incompleteness, using our ex-tensive set of image simulations (as described in Sect. 3).We counted the number of simulated galaxies with intrin-sic parameters in our selection region, and divided this bythe number of simulated galaxies with measured parametersin our selection region. Key here is that a larger region ofparameter space is simulated than we considered for the anal-ysis. This means that scatter between intrinsic and measured

1 In principle, to properly estimate the background, structures that arephysically associated with the group in question should be masked outbefore estimating the background. However, given the very low over-densities of UDGs in these groups together with the large statisticaluncertainties in this work, we find that even in an extreme case, namelywhen we mask all groups up to z ≤ 0.10, this has a negligible effect,namely at most 10% of the statistical uncertainty, on the final results(cf. Appendix A). The same is true when we estimate the backgroundfor groups using the average over all the fields. In the following wetherefore use a simple average number density per field to estimate thebackground.

A79, page 5 of 13

A&A 607, A79 (2017)

parameters can be taken into account. For this step we hadto assume an underlying intrinsic distribution in morpholog-ical parameters. We started with the values measured for thecluster population of UDGs in vdB16; a steep size distribu-tion given by n[dex−1] ∝ r−3.4

eff, and a homogeneous distribu-

tion in surface brightness (in this region of parameter space).This led to correction factors ranging from 0.57 at the lowestredshift to 1.50 at z = 0.10.

– Since we only considered sources with sizes between 1′′.5and 8′′.0 arcsec, we estimated which fraction of UDGs withphysical sizes between 1.5 and 7.0 kpc we missed in thiscount. For this we again used the size distribution. There isno correction (correction factor = 1) in the range 0.044 < z <0.051, since all physical sizes are represented in that redshiftrange. It increases only slightly towards lower redshift (sincelarge UDGs are rare), but peaks at 8.3 for groups at z = 0.10.This correction factor depends on the assumed UDG size dis-tribution, which we iterated over in Sect. 4.2. In Appendix Awe show that our final results are not significantly affectedby this assumed size distribution.

– Since vdB16 were able to consider a slightly broader rangein mean surface brightness within the effective radius (24.0 ≤〈µ(r, reff)〉 ≤ 26.5 mag arcsec−2, instead of up to 26.0), we in-crease our counts by 25%, assuming that the distribution ishomogeneous in surface brightness (vdB16, and supportedby the current analysis, see Fig. 4). We note that this correc-tion was made only for the purpose of comparing our resultsto those from vdB16 and other literature studies.

4. Results

The next two subsections present our main results, namely theabundance of UDGs as a function of halo mass, and their sizedistribution in the GAMA groups. They are measured iteratively,since the result of each of the two measurements potentially in-fluences the other. We explain the reason for this interdepen-dence in Appendix A, and conclude there that each measurementis robust and only mildly influenced by the other (i.e. to a degreethat is smaller than the size of the statistical uncertainties).

4.1. Abundance of UDGs in galaxy groups

Figure 3 shows the abundance of UDGs as a function of halomass for the GAMA groups. Group masses are based on aweak-lensing calibration in bins of their total r-band lumi-nosity (Viola et al. 2015). Two types of error bars are shown,namely Poisson uncertainties and bootstrap errors where groupsin each bin have been resampled (drawing with replacement).Data points are also presented in Table 2. We note that the sta-tistical background subtraction may in principle yield negativevalues for the abundance of UDGs in groups; in this case, for thebin with few low-mass systems that have a low overdensity com-pared to the background. In Fig. 3 we also show the measure-ments from vdB16, which were obtained from a similar analy-sis, but are based on individual clusters. We note that halo massesare determined differently between vdB16 and the current study.vdB16 measured cluster masses dynamically (cf. Sifón et al.2015), while our group masses are based on their total r-bandluminosity. However, the group masses are calibrated directlyusing weak gravitational lensing (Viola et al. 2015), while thedynamical masses of the clusters studied in vdB16 are tightlycorrelated with lensing mass (Herbonnet et al., in prep.). In Ap-pendix A we perform a conservative test to ensure that residual

Fig. 3. Abundance of UDGs as a function of halo mass. Thick blackpoints: averages in bins of halo mass. Two types of error are shown:thick errors with hats are Poisson uncertainties, while thin errors showbootstrap errors after resampling the groups in each bin. Thick redpoints: values measured for eight individual clusters in vdB16. Blacklines: best-fitting power-law relation including all data points shownhere. Dashes: relations corresponding to the ±1σ uncertainties on thepower-law index. The numbers on the top axis indicate the number ofgroups in each bin.

Table 2. Average number of UDGs per group in bins of halo mass.

Binmean N groups UDGs per group Richnesslog10[M200/M] Mr < M∗r + 2.5

12.23 8 −0.21+0.51+0.52−0.45−0.61 1.80+0.50+0.40

−0.44−0.7812.62 38 0.32+0.62+1.08

−0.55−0.93 2.93+0.30+0.39−0.27−0.40

13.03 139 1.38+0.56+1.26−0.49−1.04 4.69+0.20+0.30

−0.18−0.2613.36 107 3.31+0.89+1.51

−0.78−1.62 7.98+0.31+0.45−0.27−0.48

13.77 27 7.54+2.52+5.22−2.21−3.97 19.31+0.94+1.97

−0.83−1.5914.04 3 44.93+13.95+21.54

−12.23−38.98 39.60+3.93+5.22−3.44−7.79

14.52 1 64.29+18.16+18.16−15.92−15.92 81.84+9.66+9.66

−8.47−8.47

Notes. Bins have a width of 0.4 dex in halo mass, starting at 1012 M.These are the data points shown in Fig. 3. Two types of uncertaintyare given: first the Poisson uncertainty on the average value in eachbin, second the bootstrap uncertainty from resampling groups in eachbin (drawing with replacement). The latter uncertainty also includes thePoisson uncertainty on individual groups, and is therefore never smallerthan the former (and they are equal when there is only one group in abin). The last column lists as a comparison the richness of bright galax-ies in the same groups and bins.

uncertainties in mass calibration between the two studies do notaffect our results.

We fitted2 a power-law relation between the halo massand UDG abundance, combining both the cluster measurementsfrom vdB16, and the binned group abundances presented here.For the clusters we took errors on individual halo masses into ac-count, while for the KiDS+GAMA points we used the Poissonuncertainties, and we assumed an uncertainty of 0.1 dex in mean

2 We used a χ2 fitting recipe on data with errors in both coordinates, asdescribed in Sect. 15.3 of Press et al. (1992).

A79, page 6 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Fig. 4. Measured size distribution of galaxies in the central projectedR200 for the GAMA groups. Grey pluses: average raw counts per group.Grey note: average values expected from fore- and background interlop-ers. Grey diamond: background-subtracted values. Black data points:background-subtracted values, corrected for incompleteness, includingbootstrap resampling errors. Blue and red points: measured sizes forgalaxies split over two bins of mean effective surface brightness, also af-ter correction. In this size range, the data are well described by a power-law index of −2.71 ± 0.33 (in bins of equal logarithmic size). Greydashed line: size distribution with power-law index of −3.40, which wefound for UDGs in clusters (vdB16), arbitrarily normalised.

halo mass for each bin while performing the fit. A power law ofthe form

NUDG = (19 ± 2) ·[

M200

1014 M

]1.11±0.07

(1)

provides a good description of the data points, with χ2/d.o.f. =16.0/13.

We note that as is customary when measuring the number ofgalaxies in a halo (e.g. richness), the numbers we quote are mea-sured in a cylinder with radius R200. For the number of galax-ies within a spherical volume with radius r200, there is a depro-jection factor to be considered. Assuming that galaxies to firstorder (but see van der Burg et al. 2015) trace a Navarro-Frenk-White (NFW) profile (Navarro et al. 1997), the deprojection fac-tor depends on the concentration of the halo. The typical con-centration is a function of halo mass (e.g. Duffy et al. 2008;Dutton & Macciò 2014), so that the deprojection factor rangesfrom about ∼0.80 for groups (assuming c200 = 6), to ∼0.75 forclusters (assuming c200 = 3). If we were to deproject all valuesand repeat the fit, it would only change the best-fit slope from1.11 to 1.10. Since we are primarily interested in the relativeabundance as a function of halo mass and not in the absolutenormalisation, we did not apply a deprojection in this work.

4.2. Size distribution of UDGs

A related measurement to the abundance as a function of halomass is the size distribution of UDGs in groups. As mentionedbefore (and explained in more detail in Appendix A), both mea-surements are slightly interdependent and have to be performediteratively. Figure 4 shows the size distribution of UDGs thatpassed our main selection criteria, but have a stricter selectionin surface brightness of 24.0 ≤ 〈µ(r, reff)〉 ≤ 25.6 mag arcsec−2,since we are highly complete in this region of parameter space

(cf. Fig. 2). Each of the 325 groups contributes to the estimatedsize distribution, but the physical range that is covered dependson the redshift of each group. Since more massive groups aregenerally found at higher redshift (where the volume probed islarger), we accounted for the fact that they contain more UDGswithin their R200 disk than lower-mass groups (cf. Fig. 3). Wetherefore scaled numbers by M1.1

200, that is, by the relation foundin Sect. 4.1.

The drop in the background-subtracted counts (diamonds) atsmall physical sizes arises because groups at higher redshift donot contribute to these bins, and given that these are massiveand therefore contain many UDGs, there is a large correction ofup to a factor ∼5 for the smallest-size bin. After correcting thenumbers and fitting a power-law distribution, we obtained a bestfit n[dex−1] ∝ r−2.71±0.33

effwith χ2/d.o.f. = 0.56. We performed

the same analysis on two subsamples with mean effective surfacebrightnesses of 24.0 ≤ 〈µ(r, reff)〉 ≤ 24.8 mag arcsec−2 (shownin red) and 24.8 ≤ 〈µ(r, reff)〉 ≤ 25.6 mag arcsec−2 (shown inblue), finding power-law indices that are identical within errors.The normalisation of each subsample is roughly 50% of the fullsample, indicating that the UDG distribution is roughly uniformas a function of mean effective surface brightness in this part ofparameter space (similar to what was found by vdB16).

5. Discussion

The measured size distribution of UDGs places powerful con-straints on models that aim to explain the formation of UDGs(e.g. Amorisco & Loeb 2016). The size distribution we find(n[dex−1] ∝ r−2.71±0.33

eff) is very steep, indicating that the largest

UDGs are rare. Thanks to the strong overdensity of the clustersstudied in vdB16, the authors were able to measure the size dis-tribution more precisely to be −3.40 ± 0.19 (see their Fig. 7).While the UDG size distribution is certainly steep, there is evi-dence (at the 2σ level) that the distribution depends on environ-ment. It would be worthwhile to measure the size distributionspecifically only in the lowest-mass groups, but we find that thisresults in such a low overdensity of UDG candidates with re-spect to the field that we cannot place tighter constraints on apossible environment dependence of the UDG size distributionin this work.

The measured abundance of UDGs as a function of halo massis perhaps even more important in constraining formation sce-narios, as it directly addresses the question of where UDGs mayhave formed. We have measured the abundance now in a consis-tent manner over three orders of magnitude in halo mass. For thelowest-mass groups that enter into the analysis, we note that theycontain on average fewer than one UDG per group. A power lawof the form N ∝ M1.11±0.07

200 provides a good description of theabundance of UDGs as a function of their host halo mass. InAppendix A we show that this measurement is robust with re-spect to the main assumptions that enter into the analysis. Wenote that the measurement presented here is more precise thanthe measurement performed by vdB16, where our measurementon a cluster sample spans only one order of magnitude in halomass. vdB16 found a power-law exponent of 0.93 ± 0.16, whichis consistent within ∼1σ with this measurement.

According to our best-fit relation, Eq. (1), roughly one infive systems like the Local Group, which has a combined massof M200 ≈ 2 × 1012 M (González et al. 2014), is expected tohost a UDG within our selection boundaries. This rises to asubstantial population of ∼200 UDGs at halo mass scales ofM200 = 1015 M.

A79, page 7 of 13

A&A 607, A79 (2017)

Fig. 5. Same as Fig. 3, but now expanding the plotted range, and show-ing literature measurements (as described in the text). Uncertaintiesshown here are the bootstrap error bars, from resampling groups withineach bin.

We note that Román & Trujillo (2017b) measured the abun-dance of UDGs in three groups and found UDGs to be moreabundant per unit halo mass in groups than in higher-mass sys-tems (see Fig. 5 for a comparison), a conclusion that is differ-ent from ours. Low-number statistics may largely explain thisapparent discrepancy. We note that many of the (especially low-mass) groups we studied in this work do not contain any UDGs,thus lowering the average to fewer than 1 UDG per group inthe low-mass regime. Including Poisson scatter, the groups stud-ied in Román & Trujillo (2017b) are fully consistent with beingdrawn from a sample comparable to ours.

There is another possible interpretation, related to the groupproperties themselves. The groups studied by Román & Trujillo(2017b) are the three Hickson Compact Groups (HCGs; Hickson1982), which overlap with the SDSS Stripe 82. The HCG se-lection does not include loose groups, and is thus not rep-resentative for the diverse population of groups. In partic-ular, constituent galaxies of HCGs have likely experiencedonly few merging events in their history (Proctor et al. 2004;Mendes de Oliveira et al. 2005). This may also have positivelyaffected the survivability of UDGs in such groups.

In Fig. 5 we also show the result from Janssens et al. (2017),who measured the abundance of UDGs in a very massive clus-ter from the Hubble Frontier Fields. In Fig. 4 of Janssens et al.(2017) and in the accompanying text (see also Fig. 6 inRomán & Trujillo 2017b), a comparison is made with other mea-surements of the UDG abundance in different systems that haverecently been presented in the literature. We include them herefor reference, noting that most of them are consistent within1σ with the extrapolation of our best-fit relation, even at highermasses. Several UDGs have also been found to be possibly asso-ciated with a local galaxy group around NGC 5473 (Merritt et al.2016). However, the surface brightnesses of these UDGs aretoo faint to be detectable with the depth of the KiDS imaging.We therefore did not include these results in Fig. 5. The differ-ence in surface brightness limits is a general caveat when com-paring results from different studies. For instance, Mihos et al.(2015) studied three UDGs with surface brightnesses that are∼2 mag fainter than we can detect in the KiDS imaging. Figure 5(Román & Trujillo 2017b; Janssens et al. 2017) only includesstudies with detection limits comparable to ours.

The power law we find here suggests that the number ofUDGs per unit of host halo mass is increasing towards higher-mass haloes. It is perhaps even more insightful to compare theabundance of UDGs to the abundance of more massive galax-ies in group- and cluster-sized haloes. Measuring the so-calledmass-richness relation has been the subject of multiple stud-ies in the past decades (Marinoni & Hudson 2002; Lin et al.2004; Rines et al. 2004; Muzzin et al. 2007; Andreon & Hurn2010). With very few exceptions (Kochanek et al. 2003;van Uitert et al. 2016), they all consistently measured a rela-tion N ∝ Mα, where α < 1.0. This is also consistent withthe notion that the stellar mass fraction increases from clus-ters to group-mass haloes (Gonzalez et al. 2013; Laganá et al.2013; Budzynski et al. 2014; van der Burg et al. 2014). In par-ticular, Viola et al. (2015) measured the relation between halomass and apparent richness for the GAMA groups. They foundM200 ∝ N1.09±0.18

FoF , which, after inverting M and N, is consistentwith the general picture, namely that lower-mass haloes per unithalo mass host more bright satellites than higher-mass haloes do.We note, however, that this is measured using all FoF membersassociated with the group. This is quite different from the waywe measured the UDG abundances. We proceed by making a di-rect comparison between the UDG abundance and the richnessof GAMA groups in the same mass bins.

To help place the measured abundances of UDGs into con-text, we measured the abundance of more massive galaxies(i.e. richness) in the same groups. To make a direct compari-son, we closely followed the analysis performed for the UDGs;we measured the richness within R200, applied the same KiDSmask, and corrected the counts by assuming that the spatial dis-tribution is uniform within R200. To reduce the background cor-rection compared to a purely photometric richness measurement,we made use of the full GAMA catalogue of all spectroscopi-cally targeted galaxies. This is a valid approach given that theGAMA catalogue is essentially 100% complete down to an r-band magnitude limit of 19.8, and does not suffer from samplingproblems of close galaxy pairs thanks to multiple visits of eachfield.

We measured richnesses down to a fixed magnitude past thecharacteristic magnitude in the r band, which is M∗r = −21.21 atz = 0.10 (Blanton et al. 2003). The spectroscopic completenesslimit of 19.8 in the r band allowed us to measure the richnessdown to M∗r +2.5 at this redshift. We k-corrected this limit basedon a passively evolving stellar population with an age of 10 Gyr,and measured the richness consistently down to the same limitat all redshifts.

We estimated the velocity dispersion σLOS of each groupbased on its M200, assuming the scaling of Evrard et al. (2008).Then we selected galaxies within 5σLOS from the group redshift,which is defined as the median redshift of all FoF members. Wesubtracted the background (i.e. galaxies that are interlopers dueto their redshift) statistically, measuring the density of galaxiesin the same redshift slice (corresponding to ±5σLOS) in the sameGAMA field (G09, G12, G15), but outside the group R200.

The data points are presented in Table 2 and are shownin Fig. 6. We fitted a power-law relation to these binned datapoints in the same way as was done for the UDG abundancemeasurements, finding a best-fit relation of

NBright = (31 ± 3) ·[

M200

1014 M

]0.78±0.05

, (2)

with χ2/d.o.f. = 4.2/5.The 5σ cut is large enough to include all galaxies that are

physically associated with the group, while it is small enough to

A79, page 8 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Fig. 6. Abundance of UDGs as a function of halo mass. Thick blackpoints with errors: same as shown in Fig. 3, and these include the clusterpoints from vdB16. Black lines: best-fitting power-law relation includ-ing all black data points shown here. Blue squares: richnesses (i.e. theabundance of massive galaxies) in the same GAMA groups, with errorbars defined in a similar way. Blue lines: best-fitting power-law relationto the richness points.

not include substantial numbers of interlopers. A cut of ±3σLOSis too strict to allow for all associated galaxies, especially forgroups with low halo masses; in this case, we find the same best-fitting normalisation of 31 ± 4, but a slightly steeper slope of0.86 ± 0.06. At the other extreme, we tested a fixed redshift cutof ∆z = 0.01. This conservative selection of group galaxies givesthe same best-fit normalisation as the ±5σ cut and a similar slopeof 0.79 ± 0.05. This shows that the richness measurement is ro-bust with respect to the exact velocity cut along the line of sight.The slope is fully consistent with almost all relations measuredby the literature studies that we listed above.

The difference in trends between UDGs and more massivesatellites (mass-richness relation), indicates that UDGs are notmerely a fixed fraction of the total galaxy population that is in-dependent of the environment. It is striking that in cluster-sizedhaloes the number of UDGs is approximately equal to the num-ber of galaxies down to Mr < M∗r + 2.5. Conversely, for groupsof M200 ' 1012 M, the bright galaxies are a factor ∼10 moreabundant than UDGs. Although the number of UDGs is compa-rable to these brighter galaxies in massive groups and clusters,we note that they still constitute only a small fraction of the totalr-band luminosity (or stellar mass) even for cluster-sized sys-tems. The total stellar mass contained in UDGs is about 1.0%of the total stellar mass in galaxies with Mr < M∗r + 2.5 forclusters. This fraction drops further when compared to the totalstellar mass (due to the contribution from fainter galaxies, andthe intra-group and intra-cluster light), and is lower for groups.

UDGs therefore seem to either form, or survive, more easilyin the most massive haloes. However, they may still simply be afixed fraction of the general population of dwarf galaxies that isindependent of the environment. There are claims that the lumi-nosity function of cluster galaxies has an upturn at low luminosi-ties compared to the field (Popesso et al. 2005), although recentmeasurements do not reproduce this upturn (De Propris et al.2013; Agulli et al. 2014). If the overall dwarf galaxy populationis relatively more abundant in cluster-sized haloes, this couldmean that UDGs simply follow the same trend. An interestingapproach would be to measure the total dwarf abundance in thesame GAMA groups for a direct comparison, but the background

correction for small and faint galaxies prevents this with the cur-rent data set (see the discussion in vdB16).

Galaxies that are currently in massive galaxy clusters haveon average spent their life in more overdense environments thangalaxies that are currently in lower-mass systems. Whether asatellite galaxy can survive in a massive halo depends on theirsubhalo mass, concentration, orbit, and mass distribution of thehost halo. While tidal forces are weaker in group-sized haloesthan in clusters, mergers with other satellite galaxies are typ-ically more common because relative velocities are lower ingroups. It is hard to predict which physical process (tidal strip-ping or mergers with other galaxies) would be dominant in de-stroying UDGs in massive haloes, but it is plausible that ellipticorbits in clusters that avoid their centres are the orbits that makethe UDGs survive longest. This may explain why UDGs are rel-atively more abundant in clusters than in groups.

A detailed comparison of the morphological properties ofUDGs in different environments may also provide clues as to thepossible formation mechanisms for UDGs. The median Sérsicindex, an often-used morphological indicator of the UDG candi-dates in the clusters studied by vdB16, is nSersic = 1.4 (this me-dian value is unchanged when we only consider UDG candidatesbrighter than the surface brightness limits reached in the presentstudy). Interestingly, after accounting for background UDG can-didates, we find that the median Sérsic index of UDGs in theGAMA groups is nSersic = 2.2, and only about 33% have a Sér-sic index consistent with 1 (within uncertainties). This meansthat if we had selected UDGs in groups with a similar morphol-ogy as the UDGs found in clusters, the abundance would riseeven more steeply towards higher-mass systems. This suggeststhat the formation mechanisms for UDGs, or their subsequentevolution, depends on their environment.

6. Summary and outlook

We have expanded the work of vdB16 and measured the abun-dance of UDGs in group-sized haloes in the KiDS and GAMAsurveys. The KiDS images are only several tenths of a magni-tude shallower than the CFHT imaging stacks used in vdB16,which ensured a consistent comparison (after applying the cor-rections we described). The main conclusions of this study canbe summarised as follows.

– The size distribution at a given surface brightness is steep,with a best-fitting power law of n [dex−1] ∝ r−2.7±0.3

eff. This is

slightly shallower, but still roughly consistent (within ∼2σ)to what has been found for UDGs in clusters by vdB16.

– We were able to constrain the abundance of UDGs down tohalo masses of M200 ∼ 1012 M. In such low-mass haloes,UDGs are rare; only about one in ten of these groups containsa UDG that we can detect. The relation between the numberof UDGs and the host halo masses, which now spans threeorders of magnitude in halo mass and is well constrained,is NUDGs(R < R200) ∝ M1.11±0.07

200 . Based on this relation, weexpect about one in five systems like the Local Group to con-tain a UDG that we can detect.

– We also measured the mass-richness relation of more mas-sive galaxies in a similar way, finding a significantlyshallower relation: NBright(R < R200) ∝ M0.78±0.05

200 . Inter-estingly, in cluster-sized haloes, the number of UDGs isapproximately equal to the number of galaxies down toMr < M∗r + 2.5. In contrast, for groups of M200 ' 1012 M,the bright galaxies are a factor ∼10 more abundant thanUDGs.

A79, page 9 of 13

A&A 607, A79 (2017)

– We find a median Sérsic index of nSersic = 2.2 for UDGs ingroups, versus nSersic = 1.4 for UDGs in clusters (vdB16).This is an interesting constraint on theoretical models thataim to explain the formation and survival of UDGs, as it mayhint at different formation mechanisms, or a different subse-quent evolution, in different environments.

The different relations between the UDG abundance and therichness of more massive galaxies versus halo mass suggest thatUDGs are either destroyed more efficiently in group-sized haloesthan in cluster-sized haloes, or that galaxy clusters may playa positive role in their creation. It may also be that the over-all dwarf galaxy population is more abundant in clusters than inlower-mass systems, and that the UDGs constitute a fixed frac-tion of this general dwarf population. More detailed modellingis required to better interpret the differences measured here.

On the observational side, progress can be made by per-forming detailed comparisons between the properties of UDGsin different environments (beyond comparing a median Sérsicindex). Unfortunately, the photometric data set we used is notideal to investigate this; except for the r band, which we usedin this analysis, the imaging in the other filters is too shallowto measure meaningful colours. This prevents us from increas-ing the contrast of UDGs against the background, and we reachan overdensity of only 29%. The Hyper Suprime-Cam survey,which recently released imaging over an overlapping sky region(Aihara et al. 2017), may be the way forward to study UDGswith deeper imaging in more detail.

Acknowledgements. We thank Annie Robin, Emanuele Daddi, and RyanJohnson for insightful discussions, and an anonymous referee for a constructivereport with valuable points to improve and clarify this manuscript. The researchleading to these results has received funding from the European Research Coun-cil under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 340519. H. Hoekstra acknowledges supportfrom the European Research Council under FP7 grant number 279396. C.H. ac-knowledges support from the European Research Council under grant number647112. H. Hildebrandt is supported by an Emmy Noether grant (No. Hi 1495/2-1) of the Deutsche Forschungsgemeinschaft. D.K. is supported by the DeutscheForschungsgemeinschaft in the framework of the TR33 “The Dark Universe”.K.K. acknowledges support by the Alexander von Humboldt Foundation. R.N.acknowledges support from the German Federal Ministry for Economic Af-fairs and Energy (BMWi) provided via DLR under project No. 50QE1103.Based on data products from observations made with ESO Telescopes at theLa Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017and 177.A-3018, and on data products produced by Target/OmegaCEN, INAF-OACN, INAF-OAPD and the KiDS production team, on behalf of the KiDSconsortium. OmegaCEN and the KiDS production team acknowledge supportby NOVA and NWO-M grants. Members of INAF-OAPD and INAF-OACNalso acknowledge the support from the Department of Physics & Astronomyof the University of Padova, and of the Department of Physics of Univ. Fed-erico II (Naples). GAMA is a joint European-Australasian project based arounda spectroscopic campaign using the Anglo-Australian Telescope. The GAMAinput catalogue is based on data taken from the Sloan Digital Sky Survey andthe UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMAregions is being obtained by a number of independent survey programmes in-cluding GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS,GMRT and ASKAP providing UV to radio coverage. GAMA is funded by theSTFC (UK), the ARC (Australia), the AAO, and the participating institutions.The GAMA website is http://www.gama-survey.org/.

ReferencesAgulli, I., Aguerri, J. A. L., Sánchez-Janssen, R., et al. 2014, MNRAS, 444, L34Aihara, H., Armstrong, R., Bickerton, S., et al. 2017, PASJ, submitted

[arXiv:1702.08449]Amorisco, N. C., & Loeb, A. 2016, MNRAS, 459, L51

Amorisco, N. C., Monachesi, A., & White, S. D. M. 2016, ArXiv e-prints[arXiv:1610.01595]

Andreon, S., & Hurn, M. A. 2010, MNRAS, 404, 1922Beasley, M. A., & Trujillo, I. 2016, ApJ, 830, 23Beasley, M. A., Romanowsky, A. J., Pota, V., et al. 2016, ApJ, 819, L20Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393Blanton, M. R., Hogg, D. W., Bahcall, N. A., et al. 2003, ApJ, 592, 819Budzynski, J. M., Koposov, S. E., McCarthy, I. G., & Belokurov, V. 2014,

MNRAS, 437, 1362Dalcanton, J. J., Spergel, D. N., Gunn, J. E., Schmidt, M., & Schneider, D. P.

1997, AJ, 114, 635Davies, L. J. M., Robotham, A. S. G., Driver, S. P., et al. 2016, MNRAS, 455,

4013de Jong, J. T. A., Verdoes Kleijn, G. A., Boxhoorn, D. R., et al. 2015, A&A, 582,

A62de Jong, J. T. A., Verdoes Kleijn, G. A., Erben, T., et al. 2017, A&A, 604, A134De Propris, R., Phillipps, S., & Bremer, M. N. 2013, MNRAS, 434, 3469Di Cintio, A., Brook, C. B., Dutton, A. A., et al. 2017, MNRAS, 466, L1Driver, S. P., Hill, D. T., Kelvin, L. S., et al. 2011, MNRAS, 413, 971Duffy, A. R., Schaye, J., Kay, S. T., & Dalla Vecchia, C. 2008, MNRAS, 390,

L64Dutton, A. A., & Macciò, A. V. 2014, MNRAS, 441, 3359Erben, T., Hildebrandt, H., Miller, L., et al. 2013, MNRAS, 433, 2545Evrard, A. E., Bialek, J., Busha, M., et al. 2008, ApJ, 672, 122Gonzalez, A. H., Sivanandam, S., Zabludoff, A. I., & Zaritsky, D. 2013, ApJ,

778, 14González, R. E., Kravtsov, A. V., & Gnedin, N. Y. 2014, ApJ, 793, 91Hickson, P. 1982, ApJ, 255, 382Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454Impey, C., Bothun, G., & Malin, D. 1988, ApJ, 330, 634Janssens, S., Abraham, R., Brodie, J., et al. 2017, ApJ, 839, L17Kadowaki, J., Zaritsky, D., & Donnerstein, R. L. 2017, ApJ, 838, L21Kochanek, C. S., White, M., Huchra, J., et al. 2003, ApJ, 585, 161Koda, J., Yagi, M., Yamanoi, H., & Komiyama, Y. 2015, ApJ, 807, L2Kuijken, K., Heymans, C., Hildebrandt, H., et al. 2015, MNRAS, 454, 3500Laganá, T. F., Martinet, N., Durret, F., et al. 2013, A&A, 555, A66Lin, Y.-T., Mohr, J. J., & Stanford, S. A. 2004, ApJ, 610, 745Liske, J., Baldry, I. K., Driver, S. P., et al. 2015, MNRAS, 452, 2087Marinoni, C., & Hudson, M. J. 2002, ApJ, 569, 101Mendes de Oliveira, C., Coelho, P., González, J. J., & Barbuy, B. 2005, AJ, 130,

55Merritt, A., van Dokkum, P., Danieli, S., et al. 2016, ApJ, 833, 168Mihos, J. C., Durrell, P. R., Ferrarese, L., et al. 2015, ApJ, 809, L21Muñoz, R. P., Eigenthaler, P., Puzia, T. H., et al. 2015, ApJ, 813, L15Muzzin, A., Yee, H. K. C., Hall, P. B., & Lin, H. 2007, ApJ, 663, 150Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493Peng, E. W., & Lim, S. 2016, ApJ, 822, L31Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266Popesso, P., Böhringer, H., Romaniello, M., & Voges, W. 2005, A&A, 433, 415Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992,

Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. (NewYork, NY, USA: Cambridge University Press)

Proctor, R. N., Forbes, D. A., Hau, G. K. T., et al. 2004, MNRAS, 349, 1381Rines, K., Geller, M. J., Diaferio, A., Kurtz, M. J., & Jarrett, T. H. 2004, AJ, 128,

1078Robotham, A. S. G., Norberg, P., Driver, S. P., et al. 2011, MNRAS, 416, 2640Román, J., & Trujillo, I. 2017a, MNRAS, 468, 703Román, J., & Trujillo, I. 2017b, MNRAS, 468, 4039Rong, Y., Guo, Q., Gao, L., et al. 2017, MNRAS, 470, 4231Sifón, C., Hoekstra, H., Cacciato, M., et al. 2015, A&A, 575, A48Sifón, C., van der Burg, R. F. J., Hoekstra, H., Muzzin, A., & Herbonnet, R.

2017, MNRAS, submitted [arXiv:1704.07847]Silk, J. 2017, ApJ, 839, L13Turner, J. A., Phillipps, S., Davies, J. I., & Disney, M. J. 1993, MNRAS, 261, 39van der Burg, R. F. J., Muzzin, A., Hoekstra, H., et al. 2014, A&A, 561, A79van der Burg, R. F. J., Hoekstra, H., Muzzin, A., et al. 2015, A&A, 577, A19van der Burg, R. F. J., Muzzin, A., & Hoekstra, H. 2016, A&A, 590, A20van Dokkum, P. G., Abraham, R., Merritt, A., et al. 2015, ApJ, 798, L45van Dokkum, P., Abraham, R., Brodie, J., et al. 2016, ApJ, 828, L6van Uitert, E., Gilbank, D. G., Hoekstra, H., et al. 2016, A&A, 586, A43Viola, M., Cacciato, M., Brouwer, M., et al. 2015, MNRAS, 452, 3529Williams, R. P., Baldry, I. K., Kelvin, L. S., et al. 2016, MNRAS, 463, 2746Yagi, M., Koda, J., Komiyama, Y., & Yamanoi, H. 2016, ApJS, 225, 11Yozin, C., & Bekki, K. 2015, MNRAS, 452, 937Zaritsky, D. 2017, MNRAS, 464, L110

A79, page 10 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Appendix A: Robustness tests

Here we discuss the interdependence of the two main measure-ments in this paper, namely the abundance of UDGs as a functionof halo mass and their size distribution, and assess the robust-ness of each individual measurement. We would expect a corre-lation between redshifts and halo masses of the galaxy groupsbecause the probed volume is larger at high-z and because ofthe higher sensitivity at low-z (since the limiting magnitude ofthe spectroscopic target selection probes further down the lumi-nosity function at low-z). This correlation is the reason for thisinterdependence. For instance, the correction factors describedin Sect. 3.2 are mainly a function of redshift, but depend on theassumed size distribution. Since Fig. 3 shows the main results inbins of mass, any correlation between halo masses and redshiftsof groups would create an uncertainty in the correction factors(or an incorrectly assumed size distribution) appear as a tilt inthe abundance versus mass relation. The good news is that al-though a correlation between redshift and halo mass exists forour sample, it is moderate in this redshift range (Fig. A.1, andfor the distribution over a larger redshift baseline, cf. Fig. 16 inRobotham et al. 2011). This means that the results do not sen-sitively depend on each other, as we illustrate with several testsbelow.

Our fiducial measurement of the abundance-mass relation as-sumes a UDG size power-law with exponent =−2.70. This leadsto a measured abundance power-law = 1.11 ± 0.07 (the mainresult presented in Sect. 4.1). When we change the assumedUDG size power-law to exponent =−3.0 or −2.4 (i.e. 1σ loweror higher than the fiducial model), we measure an abundancepower-law exponent of 1.10±0.07 and 1.13±0.07, respectively.Even when we take the steeper size measured in vdB16 (sizepower-law exponent of −3.40), we measure a moderate changein the abundance power-law exponent (1.08 ± 0.07). This showsthat the assumed UDG size distribution does not significantlyaffect the measured UDG abundance versus halo mass.

As described in the main text, we have assumed a weak-lensing calibration of the total mass associated with the galaxygroups, based on their total r-band luminosity, as outlined inViola et al. (2015). We explore what would happen if we wereto very conservatively increase or decrease the masses of thegalaxy groups by 20%. This results in a measured abundancepower-law exponent of 1.10±0.06 and 1.07±0.06, respectively.We can conclude that uncertainties on the average GAMA groupmass measurements do not affect our results.

The results presented in this work rely on a statistical back-ground correction, which is substantial for the marginally over-dense groups studied here. There are different ways of perform-ing this background correction, and as explained in the text, weused the average density of UDG candidates (with angular sizes1′′.5 ≤ reff ≤ 8′′.0) over the entire KiDS region. If we were tomeasure the density outside of the projected virial radii of allGAMA groups up to redshift z ≤ 0.10, we would measure anabundance power-law with exponent 1.09± 0.07. Our results arethus robust with respect to the details of the background subtrac-tion method.

Another sanity check we performed is to repeat the sameanalysis using the three million simulated sources that wereinjected at random positions into the KiDS images. We made25 random resamplings of the simulated catalogue, where wegave a probability for each source to be picked based on theirsizes. In this way, every resampled catalogue has the same sizedistribution as the observed size distribution of UDG candidates.Each realisation thus contains 16 595 sources (±Poisson noise).

Fig. A.1. Distribution in redshift and mass of the 325 GAMA galaxygroups we studied. The mass bins we used for the abundance measure-ments are indicated.

Fig. A.2. Abundance of UDGs as a function of halo mass. Small blackdots: individual measurements per group (using real data). Thick blackpoints and errors: identical to that shown in Fig. 3. Red points: meanUDG number densities per group when the UDGs are injected at ran-dom positions into the KiDS images, with 68% confidence interval frombootstrapping random samples. The inset shows a zoomed-in region.The abundance measurements show a clear signal, while the “randoms”are consistent with zero.

The median measurements, with 68% confidence regions, areshown by the red points in Fig. A.2. This shows that the signalis consistent with zero, with a spread between the 25 realisationsthat is roughly equal to the estimated uncertainties on the realdata points.

If bright group galaxies, or intra-group light (IGL), hadplayed a significant role in defining the sample of UDGs (in par-ticular, preventing us from detecting UDGs close to the groupcentres), this would have shown up as a deficit in the randomsamples, which is not the case here.

We note that there are three groups in the mass bin 1014.0 ≤

M200/M ≤ 1014.4, but one of them has a very marginal over-density compared to the background. This results in a very largebootstrap error.

The measured relation between the abundance and mass ofgroups in turn affects the estimated size distribution. First werecall that the smallest UDGs that are plotted in Fig. 4 are not

A79, page 11 of 13

A&A 607, A79 (2017)

directly measured in the highest-z groups because their angularsizes are then smaller than 1′′.5. The fitted power-law size distri-bution thus already enters into the correction factor. Moreover,we normalised the distribution here per “virial disk” (π · R2

200).Since the number of UDGs is a strong function of the halomass (cf. 3), primarily probing high-mass haloes at high redshiftwould introduce an artificial tilt in the measured relation. To cir-cumvent this, we scaled the UDG numbers by dividing them overM1.1

200. Again, if we were to probe the same halo mass distributionat different redshifts, such a scaling would have no effect.

We tested the effect of our assumed abundance versus halomass relation on the estimated size distribution. Our fiducialassumption was a scaling by M1.1

200, which led to a size power-law exponent of −2.71±0.33. If we were to assume an abundance

scaling of M1.3200 or M0.9

200 (i.e. 3σ higher or lower than the fiducialmodel), we find −2.70 ± 0.34 and −2.70 ± 0.33, respectively.Uncertainties in the assumed abundance-mass distribution thushave a negligible effect on the measured size distribution.

Appendix B: Examples

To provide a qualitative sense of the depth of the KiDS imaging,Fig. B.1 shows selected UDG candidates in different parts of pa-rameter space (as described in Sect. 3.1 and shown in Fig. 2).The best-fitting Sérsic parameters are listed in Table B.1. Com-pleteness limits are quantified in Sect. 3.

A79, page 12 of 13

R. F. J. van der Burg et al.: The abundance of ultra-diffuse galaxies from groups to clusters

Table B.1. Best-fitting GALFIT parameters of the examples shown in Fig. B.1.

reff 〈µ(r, reff)〉 Sérsic reff 〈µ(r, reff)〉 Sérsic reff 〈µ(r, reff)〉 Sérsic[arcsec] [mag arcsec−2] n [arcsec] [mag arcsec−2] n [arcsec] [mag arcsec−2] n

(a) 7.31 24.10 0.96 (i) 5.41 24.93 1.04 (q) 2.74 25.42 1.95(b) 5.75 24.26 2.11 (j) 4.12 24.69 2.86 (r) 1.61 25.39 3.09(c) 4.67 24.49 2.63 (k) 2.62 24.53 1.94 (s) 6.77 25.67 0.87(d) 3.62 24.44 2.00 (l) 1.73 24.50 2.29 (t) 6.00 25.56 1.83(e) 3.35 24.23 1.50 (m) 6.86 25.26 3.98 (u) 4.88 25.78 2.53(f) 1.75 24.13 1.10 (n) 5.54 25.01 0.75 (v) 3.70 25.73 2.34(g) 6.72 24.58 3.43 (o) 4.71 25.06 0.80 (w) 2.72 25.76 2.71(h) 6.50 24.97 3.45 (p) 3.72 25.00 2.74 (x) 1.95 25.70 1.14

Fig. B.1. Example of typical galaxies we selected in different parts of parameter space (angular size versus mean surface brightness within theeffective radius). Left panels: original r-band images, spanning 21′′ × 21′′ centred on the galaxy. Right panels: residual of the r-band image aftermodel subtraction. The best-fit morphological parameters are listed in Table B.1.

A79, page 13 of 13